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A model of ideal elastomeric gels for polyelectrolyte gels Cite this: DOI: 10.1039/c3sm52751d
Jianyu Li,a Zhigang Suoab and Joost J. Vlassak*a The concept of the ideal elastomeric gel is extended to polyelectrolyte gels and verified using a polyacrylamide-co-acrylic acid hydrogel as a model material system. A comparison between mixing and ion osmosis shows that the mixing osmosis is larger than the ion osmosis for small swelling ratios, while the ion osmosis dominates for large swelling ratios. We show further that the non-Gaussian chain effect Received 30th October 2013 Accepted 6th January 2014
becomes important in the elasticity of the polymer network at the very large swelling ratios that may occur under certain conditions of pH and salinity. We demonstrate that the Gent model captures the
DOI: 10.1039/c3sm52751d
non-Gaussian chain effect well and that it provides a good description of the free energy associated with
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the stretching of the network. The model of ideal elastomeric gels fits the experimental data very well.
1. Introduction A polyelectrolyte gel is formed when a cross-linked polymer network carrying ionizable groups, generally acidic, absorbs a solvent containing ionic species. Polyelectrolyte gels have the capacity to absorb large amounts of solvent, leading to their use in a broad range of applications in healthcare and personal hygiene.1 The degree of swelling depends sensitively on environmental conditions such as pH and salinity,2–4 and in some cases even electric elds,5 temperature,6 or light.7 Thus polyelectrolyte gels can also be used as smart materials in sensors and actuators.8,9 These applications have motivated the development of theoretical models characterizing the chemo-mechanical behavior of polyelectrolyte gels. Katchalsky et al. pioneered the study of polyelectrolyte solutions and gels.10,11 Tanaka et al. validated the use of Donnan's theory for the osmotic pressure that develops when mixing polymer chains and ions.12 A scaling theory proposed by de Gennes et al. and further developed by Rubinstein et al. was used to predict the behavior of polyelectrolyte gels as a function of chain length, the type of cross-link, monomers, and salt concentrations.13–15 Several theoretical models for the swelling behavior of polyelectrolyte gels based on the Flory–Rehner theory have also been reported.16,17 Less attention has been paid to direct comparisons between experiments and theoretical predictions of a comprehensive eld theory. A notable exception is the work by Prudnikova and Utz,18 who applied the thermodynamic eld theory by Hong et al.,19 to predict the Donnan potentials, but found a systematic error between theoretical predictions and experiments. Evidently, a comprehensive quantitative model for the complex chemomechanical behavior of polyelectrolyte gels is still elusive. a
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA. E-mail: [email protected]
b
Kavli Institute for Bionano Science & Technology, Harvard University, Cambridge, MA 02138, USA
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One possible approach is to develop a model based on the concept of the ideal elastomeric gel, which was proposed by Cai and Suo,20 and veried recently for polyacrylamide hydrogels in an experimental study.21 This study showed that stresses applied under various loading conditions are balanced by the elasticity of the polyacrylamide network and the osmotic pressure. It was further demonstrated that the presence of cross-links does not affect the mixing energy of the polymer chains and the solvent measurably. An important feature of the ideal elastomeric gel concept is the absence of a specic statistical model for the behavior of the polymer network or a specic expression for the energy of mixing; a full model of a gel can then be realized through a series of simple experiments. Since the theoretical framework of the ideal elastomeric gel model is quite general, the same approach can also be applied for other types of gels. In this paper, we apply the concept of the ideal elastomeric gel to polyelectrolyte gels using a polyacrylamide-co-acrylic acid hydrogel as a model material system. We demonstrate experimentally that in polyelectrolyte gels the elasticity of the network is balanced by the osmotic pressure associated with two distinct processes: mixing of the polymer and solvent and the redistribution of the ionic species. In the present model, the contribution of the ionic species to the osmotic pressure is quantied using Donnan's theory. A comparison between the mixing osmosis and the ion osmosis shows that the mixing osmosis is larger than the ion osmosis for small swelling ratios, while the ion osmosis dominates for very large swelling ratios. We show further that the non-Gaussian chain effect becomes important in the elasticity of the polymer network at the very large swelling ratios that may occur under certain conditions of pH and salinity. We demonstrate that the Gent model captures the nonGaussian chain effect well and that it provides a good description of the free energy associated with the stretching of the network.
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2.
Paper
Theory
Following our previous work,20,21 we derive the equations of state for a polyelectrolyte gel. To focus on the main ideas, we consider deformation in the principal directions, although the theory is readily expanded to the more general case.22 In the reference state, the gel is a unit cube of a dry polymer, containing no solvent and subject to no applied forces (Fig. 1a). In the current state, submerged in a solution and subject to applied forces, the gel absorbs solvent molecules and various ionic species, and stretches into a rectangular block of dimensions l1, l2 and l3 (Fig. 1a). The volume ratio of gel to dry polymer, J, known as the swelling ratio, relates to the dimensions of the rectangular block by J ¼ l1l2l3. The gel and the external solution contain four mobile species: solvent molecules, protons, counterions with charge opposite to the charge xed on the polymer chains, and co-ions with charge of the same sign as the xed charge (Fig. 1b). We dene the nominal concentration Ca of species a as the number of species a in the current state divided by the volume of the dry network. The same number divided by the volume of the gel in the current state denes the true concentration ca. The two quantities are evidently related by Ca ¼ ca J. As part of the swelling process, the acidic groups AH on the polymer chains of the network may dissociate, leaving xed charge A on the chains. This dissociation reaction, AH 4 A + H+, reaches equilibrium when CHþ CA Na Ka ¼ CAH J
(1)
where Na is Avogadro's constant and Ka is the acid dissociation constant.
Fig. 1 (a) A unit block of the polyelectrolyte gel is submerged in a solution environment containing various ionic species. The inset is the structural formula of polyacrylamide-co-acrylic acid. (b) The polyelectrolyte gel, composed of cross-linked polymer chains with fixed charge and acidic groups, absorbs solvent and mobile ions, while the external solution contains solvent, protons, co-ions and counterions.
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We recall the two basic assumptions underlying the concept of an ideal elastomeric gel. (1) Molecular incompressibility: the volume of the ideal elastomeric gel is equal to the sum of the volume of the dry network and that of the solvent, J ¼ 1 + UsCs,
(2)
where Us is the volume per solvent molecule s. In the case of polyelectrolyte gels, we assume further that the concentrations of the ionic species are generally much lower than those of the solvent species,17 so that eqn (2) remains valid. (2) Separability of the Helmholtz free energy: the Helmholtz free energy of the ideal elastomeric gel consists of one term that represents the elastic stretching of the polymer network and one term that is associated with mixing of the polymer and solvent. In the case of polyelectrolyte gels, the xed and mobile ions also contribute to the Helmholtz free energy. Following prior work,12,16,17 we formulate the total Helmholtz free energy referred to a unit volume in the reference state as the sum of the following contributions: W ¼ Wstretch + Wmix + Wion + Wdis,
(3)
where Wstretch is the energy contribution due to the stretching of the network. This term is a function of the stretches (l1, l2, l3) and depends on the cross-link density of the network. The term Wmix is the contribution associated with the mixing of the polymer and solvent. This term is a function of the swelling ratio, J, given by eqn (2) and is taken independent of the crosslink density. The term Wion is associated with the mixing of mobile ions and generally takes the form Wion ¼ Wion(J, CH+, C+, C), where the subscripts + and – refer to the counterions and co-ions, respectively. If the concentrations of the ions are low and the polymer network is weakly charged, the electrostatic interactions between the ions in the gel are negligible,12 and Wion is determined mainly by the entropy of mixing of the mobile ions: ! " ! C þ Cþ Wion ¼ kT CHþ log refH 1 þ Cþ log ref 1 cþ J cHþ J # C þ C log ref 1 ; (4) c J where cref refers to the standard concentration, dened as 1 mol L1. For strongly charged polyelectrolyte gels, the energy contribution due to electrostatic interactions between mobile ions and xed charge may be signicant and needs to be considered.11,23 The electrostatic effect can be readily applied to this theoretical framework, but is not further considered in this study. The last term in eqn (3), Wdis, denotes the change in free energy caused by dissociation of the acidic groups on the polymer chains. This term contains both an entropic and an enthalpic term, CA Wdis ¼ kT CA log CA þ CAH CAH þ gCA ; (5) þ CAH log CA þ CAH
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where g is the increase in enthalpy when an acidic group dissociates. Under the constraints of mass conservation and electroneutrality, CA and CAH are readily expressed in terms of CH+, C+ and C: CA ¼ CH+ + C+ C
(6)
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and CAH ¼ f/v (CH+ + C+ C),
(7)
where f is the number of acidic groups attached to the network divided by the total number of monomers forming the network and v is the volume per monomer. Consequently, the total Helmholtz free energy can be written as a function of six independent variables: the stretches (l1, l2, l3) and the concentrations of the protons, counterions, and co-ions (CH+, C, C+). When a dry polymer network is submerged in a solution and subjected to forces, the network absorbs the mobile species and deforms, eventually attaining a state of thermodynamic equilibrium. The condition for equilibrium can be formulated as follows. We dene the stresses s1, s2 and s3 as the applied forces divided by the areas of the faces of the block of the gel in the current state (Fig. 1a). Thus, the forces on the faces of the rectangular block are s1l2l3, s2l3l1 and s3l1l2. Associated with a small change in the dimensions of the gel, these forces perform work on the gel s1l2l3dl1 + s2l3l1dl2 + s3l1l2dl3. The gel and the solution can also exchange four mobile species: solvent molecules, protons, counterions and co-ions. Let ma be the chemical potential of species a—since the gel and the solution are in a state of equilibrium, the chemical potential of each species is uniform in the system. When a number dCa (per unit volume of dry gel) of species a are transferred from the solution to the gel, the free energy of the gel changes by madCa. This statement holds for solvent molecules, counterions, and co-ions but not for protons because the number of protons in the gel can also change by dissociation of the acidic groups on the polymer chains. Consequently, in equilibrium the total Helmholtz free energy W obeys dW ¼ s1l2l3dl1 + s2l3l1dl2 + s3l1l2dl3 + msdCs + mH+(dCH+ dCA) + mdC + m+dC+. Combining eqn (2)–(8) yields the following expression, " # vWstretch ðs1 þ Pmix þ Pion Þl2 l3 dl1 vl1 " # vWstretch þ ðs2 þ Pmix þ Pion Þl3 l1 dl2 vl2 " # vWstretch þ ðs3 þ Pmix þ Pion Þl1 l2 dl3 vl3 " # vW þ ðmþ mHþ Þ dCþ vCþ " # vW vW þ ðm þ mHþ Þ dC þ dCHþ ¼ 0 vC vCHþ
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(8)
(9)
where dWmix ðJÞ ; dJ
(10)
vWion ðJ; CHþ ; Cþ ; C Þ ms þ : vJ Us
(11)
Pmix ¼ Pion ¼
here Pmix is the osmotic pressure in the gel in equilibrium with the pure solvent, while Pion represents the osmotic pressure in the gel due to the mobile ions; ms/Us is the osmotic pressure in the gel due to the chemical potential of the solvent in the external solution containing ionic species: ms ¼ kTUs c*Hþ þ c*þ þ c* (12) where c*a is the true concentration of species a in the external solution.17 Substituting eqn (4) and (12) into (11) yields (13) Pion ¼ kT cHþ þ cþ þ c c*Hþ c*þ c* : The network, the solvent and the applied forces are in equilibrium, if eqn (9) holds for arbitrary and independent small changes in the three stretches (l1, l2 and l3) and the three nominal concentrations (C+, C, CH+). Consequently, the expressions in the brackets in front of each of the six terms in eqn (9) must vanish individually, yielding the following equations: vWstretch s1 ¼ Pmix Pion ; (14a) l2 l3 vl1 s2 ¼
vWstretch Pmix Pion ; l1 l3 vl2
(14b)
s3 ¼
vWstretch Pmix Pion : l1 l2 vl3
(14c)
vW ¼ mþ mHþ : vCþ
(15a)
vW ¼ m þ mHþ : vC
(15b)
vW ¼ 0: vCHþ
(16)
The applied stresses are balanced by the elasticity of the network, the osmotic pressure due to mixing of the polymer and solvent, and the osmotic pressure due to mixing of ions. Eqn (14) expresses mechanical equilibrium; eqn (15) provides the equilibrium conditions with respect to the exchange of counterions and co-ions between the gel and its surrounding; eqn (16) formulates the equilibrium condition for dissociation of the acidic groups on the polymer chains. Substituting eqn (3)– (5) into (16) leads to g NA Ka ¼ cref : (17) Hþ exp kT Eqn (2), (14)–(16) constitute the equations of state of the polyelectrolyte gel, relating the thermodynamic variables: l1, l2, l3, s1, s2, s3, C+, C, CH+. Once the free energy contributions due to stretching of the network, Wstretch, and the osmotic pressure,
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Pion, have been characterized experimentally, the equations of state provide a complete description of the chemo-mechanical behavior of the polyelectrolyte gel. The energy contribution due to stretching is readily determined from stress–strain measurements. Characterizing Pion is less straightforward since it requires knowledge of the concentrations (CH+, C+, C) inside the gel, which are difficult to measure in practice.24 Instead, we adopt Donnan's theory,17 which provides a relationship between the ion concentrations inside the gel and those in the external solution, 1 cþ cHþ c ¼ ¼ * : (18) c*þ c*Hþ c Tanaka et al. successfully applied Donnan's theory to capture the swelling behavior of an ionic gel quantitatively.12 The concentrations of the three species inside the gel also satisfy the equilibrium condition (1) for dissociation of the acidic groups, yielding cHþ ðcHþ þ cþ c Þ ¼ NA K a : f ðcHþ þ cþ c Þ vJ
(19)
If f, J, c*Hþ , c*+, c*, and Ka are known, the concentrations (cH+, c+, c) of the ionic species in the gel are solved readily from eqn (18) and (19).
3.
Experimental methods
To determine the free energy contributions of stretching and the osmotic pressure in the gels in equilibrium with the pure solvent, and to verify the equations of state for polyelectrolyte gels, a series of polyacrylamide-co-acrylic acid gels were synthesized according to the following protocol;12 a summary of the composition and designation of each sample is provided in Table 1. Acrylamide (AAM), acrylic acid (AA), N,N0 -methylenebis(acrylamide) (MBAA) and ammonium persulfate (APS) were acquired from Sigma Aldrich. The overall monomer (AAM + AA) concentration was 1000 mM. As indicated in Table 1, the mole fraction of acrylic acid, f, was varied systematically, but kept small to ensure a low density of acidic groups on the polymer chains. Different cross-link densities were obtained by varying the weight percentage of MBAA (Table 1). APS (7 mM) was added as an initiator for the free-radical polymerization. The chemicals (AAM, AA, MBAA and APS) were mixed using 10 mL syringes. Part of the mixture (6 mL) was poured into a glass mold of 1 mm thickness. The rest of the solution (4 mL) was kept in the syringe to fabricate samples with a cylindrical shape. Gelation occurred over a period of one hour on a hot plate at 70 C. To prevent hydrolysis of acrylamide and to precisely control the fraction of acidic groups, we did not use the usual accelerator, tetramethyl-ethylenediamine (TEMED). Aer gelation, the samples were stored at room temperature for 24 hours, and then transferred to a large body of distilled water to remove unreactive chemicals and other impurities, while the gels were allowed to swell freely to equilibrium. This step lasted for
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Table 1 Composition and parameters of polyacrylamide-co-acrylic acid gelsa
Notation
MBAA (wt%)
f
J0
NkT (kPa)
AA01-080 AA02-080 AA04-080 AA01-216 AA02-216 AA04-216 AA01-433 AA02-433 AA04-433
0.80 0.80 0.80 2.16 2.16 2.16 4.33 4.33 4.33
0.01 0.02 0.04 0.01 0.02 0.04 0.01 0.02 0.04
140 590 654 71 79 88 36 37 61
2.33 3.62 3.71 10.83 10.22 10.07 15.96 22.22 17.39
a f is the mole fraction of AA versus (AA+AAM), J0 is the equilibrium swelling ratio of the gel in distilled water, and NkT is the crosslink density.
around 10 days, during which fresh distilled water was added every 24 hours. The mass of the fully swollen cylindrical samples (mgel) was measured using an analytical scale with an accuracy of 10 mg. The samples were subsequently frozen at a temperature of 80 C and transferred to a freeze–dry system (Labconco Corporation) with a collector temperature of 50 C and a vapor pressure of 0.040 mbar. This dehydration process took three days to complete, aer which the mass of the dehydrated samples (mdry) was measured. Swelling ratios were then calculated using
mgel mdry rwater J ¼1þ : (20) mdry =rdry where rwater is the density of water taken as 1.000 g cm3 and rdry is the density of the dry polymer network taken as the density of polyacrylamide at 1.443 g cm3.25 The swelling ratios, J0, in distilled water are listed in Table 1 for each of the samples. Swelling ratios in subsequent experiments were calculated from J0 and the change in diameter, J ¼ J0(Df/Di)3, where Di and Df are the initial and nal diameters of the sample. The responses of the gels to changes in pH and salinity of the external solution were characterized using samples AA04-080 and AA04-433. Small gel cylinders of diameters 8 or 12 mm were punched out of fully swollen gel sheets. These samples were transferred to solutions of pH varying from 2.5 to 12. Acidic solutions (pH < 7) were prepared by adding HCl to distilled water, while KOH was used for alkaline solutions (pH > 7). Gel cylinders were also transferred to solutions with NaCl concentrations varying from 107 M to 102 M. The use of buffer solutions was avoided to not introduce additional electrochemically active species. The external solutions were renewed every 24 h. We recorded the diameters of the cylinders using a pair of calipers before and aer reaching the new equilibrium state. Small gel cylinders with 8 mm diameter were used for the mechanical tests. A series of uniaxial compression tests were performed using an AR-G2 rheometer (TA Instruments). The rheometer platen approached the samples at a speed of 5 mm This journal is © The Royal Society of Chemistry 2014
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s1 until a rise in force was detected, indicating contact between the platen and the gel. Upon the contact, the displacement rate of the platen was increased to 20 mm s1 until a pre-determined stretch ratio, typically 0.8–0.9, was achieved. The entire loading process was nished within 2 minutes to avoid any redistribution of solvent inside the gel and thus maintain incompressibility. Both force and displacement were recorded continuously throughout the experiment.
4. Results and discussion Mechanical characterization Fig. 2a shows the swelling ratios in distilled water for each of the samples. Typical values are well in excess of 50 so that the water content of the most swollen gels is greater than 95%. As illustrated in the gure, the swelling ratio changes rapidly with the fraction of acidic groups and the cross-link density. Even in distilled water, weakly charged and loosely cross-linked gels (e.g. AA02-080 and AA04-080) can swell to more than 500 times their dry volume. The contribution of the free energy of stretching, Wstretch, was determined from uniaxial compression tests on samples fully swollen in distilled water. During the tests, the gels behaved like incompressible solids (l1l2l3 ¼ l03), because there was insufficient time for diffusion. We then used the Gaussian chain model, " 2 # 2 NkT l3 l1 s3 s1 ¼ ; (21a) l0 l0 l0 NkT s3 s2 ¼ l0
" 2 # 2 l3 l2 ; l0 l0
(21b)
to t the experimental stress–stretch relations, where l0 ¼ J01/3 is the stretch ratio aer swelling in distilled water, N is the number of polymer chains per unit volume of the dry network, k is Boltzmann's constant, and T is the absolute temperature. Considering uniaxial compression along the x3-direction, these equations reduce to a single equation NkT 2 1 l ; s3 ¼ (22) l0 l
where l ¼ l3/l0 is the stretch relative to the freely swollen state. It is evident from eqn (22) that NkT/l0 represents the shear modulus of the gels, while also allowing us to determine the cross-link density (N) from the initial slope of the stress–stretch curve. In our previous work on polyacrylamide gels, the Gaussian-chain model provided an excellent description of the mechanical behavior of the gels. As expected, the Gaussianchain model also provides a very good t for the experimental stress–stretch curves for the polyacrylamide-co-acrylic acid gels. Since the gels were swollen in distilled water, they had moderate swelling ratios and did not exhibit any non-Gaussian behavior. Fig. 2b illustrates that NkT increases with increased amounts of cross-linker, but is independent of the fraction of acrylic acid, f. A correlation between the modulus and the ionic components inside the gel has been reported in the literature,13 a phenomenon attributed to the electrostatic interaction between the xed ions on the polymer chain. In this study, the fraction of acidic groups is intentionally maintained at a level below 5%, so that xed ions are on average separated by at least 20 neutral segments resulting in a weak electrostatic interaction between the ions.
Characterization of the osmotic pressure terms The osmotic pressure Pion of the gel samples can be calculated by combining eqn (13) with eqn (18) and (19). The results for an external solution of distilled water with a slightly acidic pH of 6.5 are plotted as a function of f/J in Fig. 3a – data for gels with different cross-link densities and acid group contents are shown, but they collapse into a single master curve because the ionic osmotic pressure is a unique function of the current concentration of acid groups, f/J, and the pH of the external solution. The ionic osmotic pressure is shown to increase with acid group content and decrease with decreasing cross-link density. The former is caused by the higher concentration of active species and the latter occurs because the equilibrium swelling ratio is larger, which results in a lower concentration of active species. Using the equations of state, it is now possible to derive the function Pmix(J) from our experiments without assuming a specic model for Wmix. The stress–strain curves suggest that Wstretch is well described by the Gaussian-chain model, i.e.,
Fig. 2 (a) The equilibrium swelling ratio of the gels fully swollen in distilled water (J0) and (b) the crosslink density (NkT) of the gels, obtained from the stress–stretch measurements, is plotted versus the fraction of acidic groups (f) and the crosslinker input (MBAA%).
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Fig. 3 (a) The calculated osmotic pressure due to ions (Pion) versus the density of acidic groups (f/J), and (b) the osmotic pressure due to mixing (Pmix) as a function of the swelling ratio (J), measured from gels fully swollen in distilled water. The same color implies the same input of crosslinker (MBAA) in the synthesis, while the fraction of acidic groups (f) is varied. The data points marked by PAAM data were extracted from the measurement on polyacrylamide gels.21
1 Wstretch ¼ NkT l1 2 þ l2 2 þ l3 2 3 2 logðl1 l2 l3 Þ : 2
(23)
Under free-swelling conditions, the stretches are isotropic, l1 ¼ l2 ¼ l3 ¼ J01/3 and all applied stress components vanish, s1 ¼ s2 ¼ s3 ¼ 0. Substituting into eqn (14) yields NkT 2 Pmix ¼ J0 3 1 Pion ; (24) J0 from which Pmix can be obtained directly once J0, NkT, and Pion have been determined experimentally. Fig. 3b shows the plot of the function, Pmix, as a function of swelling ratio for gels with different cross-link densities and acid contents. We have also included Pmix measured for polyacrylamide (PAAM) hydrogels in our previous work.21 Regardless of the hydrogel cross-link density or acid content, Pmix collapses into a single master curve over a large range of swelling ratios, i.e., Pmix is a function of a single variable, J. This is a direct consequence of the low concentration of acid groups in the hydrogels, which makes the overall chemistry of the polymer chains close to that of pure polyacrylamide. The responses of the gels to changes in pH or salinity of the external solution were also characterized. Fig. 4a shows the plot of the swelling ratio as a function of pH, along with experimental data obtained by Tanaka et al.12 A qualitative understanding of the trends in the data is as follows. When the pH of
the external solution is low, protons are abundant and the acidic groups on the polymer chains are fully protonated. The gel behaves like a neutral gel with a relatively small swelling ratio because of the absence of an ionic osmotic pressure Pion. As the pH approaches 8, the acidic groups dissociate and create xed charge on the polymer chains. An imbalance of mobile ions builds up between the gel and the external solution, and the increasing Pion makes the gel swell. Once the pH increases further, the acidic groups are fully dissociated and the ionic osmotic pressure Pion drops because of the abundant presence of mobile ions everywhere. The response to the changes in salinity, the concentration of NaCl in the external solution, is shown in Fig. 4b; an understanding similar to that for Fig. 4a applies here. The gel acts as an ion exchanger. As the concentration of NaCl increases, protons in the gel are exchanged with Na+ ions and the acidic groups continue to dissociate. At approximately 104 M of NaCl, the acidic groups are completely dissociated. The excess of NaCl reduces the ion imbalance between the gel and the solution, and the degree of swelling decreases. In the model we assumed an ideal ionic solution and insignicant effects due to electrostatic interactions between xed and mobile ions, both valid for low concentrations (
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A model of ideal elastomeric gels for polyelectrolyte gels Cite this: DOI: 10.1039/c3sm52751d
Jianyu Li,a Zhigang Suoab and Joost J. Vlassak*a The concept of the ideal elastomeric gel is extended to polyelectrolyte gels and verified using a polyacrylamide-co-acrylic acid hydrogel as a model material system. A comparison between mixing and ion osmosis shows that the mixing osmosis is larger than the ion osmosis for small swelling ratios, while the ion osmosis dominates for large swelling ratios. We show further that the non-Gaussian chain effect Received 30th October 2013 Accepted 6th January 2014
becomes important in the elasticity of the polymer network at the very large swelling ratios that may occur under certain conditions of pH and salinity. We demonstrate that the Gent model captures the
DOI: 10.1039/c3sm52751d
non-Gaussian chain effect well and that it provides a good description of the free energy associated with
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the stretching of the network. The model of ideal elastomeric gels fits the experimental data very well.
1. Introduction A polyelectrolyte gel is formed when a cross-linked polymer network carrying ionizable groups, generally acidic, absorbs a solvent containing ionic species. Polyelectrolyte gels have the capacity to absorb large amounts of solvent, leading to their use in a broad range of applications in healthcare and personal hygiene.1 The degree of swelling depends sensitively on environmental conditions such as pH and salinity,2–4 and in some cases even electric elds,5 temperature,6 or light.7 Thus polyelectrolyte gels can also be used as smart materials in sensors and actuators.8,9 These applications have motivated the development of theoretical models characterizing the chemo-mechanical behavior of polyelectrolyte gels. Katchalsky et al. pioneered the study of polyelectrolyte solutions and gels.10,11 Tanaka et al. validated the use of Donnan's theory for the osmotic pressure that develops when mixing polymer chains and ions.12 A scaling theory proposed by de Gennes et al. and further developed by Rubinstein et al. was used to predict the behavior of polyelectrolyte gels as a function of chain length, the type of cross-link, monomers, and salt concentrations.13–15 Several theoretical models for the swelling behavior of polyelectrolyte gels based on the Flory–Rehner theory have also been reported.16,17 Less attention has been paid to direct comparisons between experiments and theoretical predictions of a comprehensive eld theory. A notable exception is the work by Prudnikova and Utz,18 who applied the thermodynamic eld theory by Hong et al.,19 to predict the Donnan potentials, but found a systematic error between theoretical predictions and experiments. Evidently, a comprehensive quantitative model for the complex chemomechanical behavior of polyelectrolyte gels is still elusive. a
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA. E-mail: [email protected]
b
Kavli Institute for Bionano Science & Technology, Harvard University, Cambridge, MA 02138, USA
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One possible approach is to develop a model based on the concept of the ideal elastomeric gel, which was proposed by Cai and Suo,20 and veried recently for polyacrylamide hydrogels in an experimental study.21 This study showed that stresses applied under various loading conditions are balanced by the elasticity of the polyacrylamide network and the osmotic pressure. It was further demonstrated that the presence of cross-links does not affect the mixing energy of the polymer chains and the solvent measurably. An important feature of the ideal elastomeric gel concept is the absence of a specic statistical model for the behavior of the polymer network or a specic expression for the energy of mixing; a full model of a gel can then be realized through a series of simple experiments. Since the theoretical framework of the ideal elastomeric gel model is quite general, the same approach can also be applied for other types of gels. In this paper, we apply the concept of the ideal elastomeric gel to polyelectrolyte gels using a polyacrylamide-co-acrylic acid hydrogel as a model material system. We demonstrate experimentally that in polyelectrolyte gels the elasticity of the network is balanced by the osmotic pressure associated with two distinct processes: mixing of the polymer and solvent and the redistribution of the ionic species. In the present model, the contribution of the ionic species to the osmotic pressure is quantied using Donnan's theory. A comparison between the mixing osmosis and the ion osmosis shows that the mixing osmosis is larger than the ion osmosis for small swelling ratios, while the ion osmosis dominates for very large swelling ratios. We show further that the non-Gaussian chain effect becomes important in the elasticity of the polymer network at the very large swelling ratios that may occur under certain conditions of pH and salinity. We demonstrate that the Gent model captures the nonGaussian chain effect well and that it provides a good description of the free energy associated with the stretching of the network.
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Theory
Following our previous work,20,21 we derive the equations of state for a polyelectrolyte gel. To focus on the main ideas, we consider deformation in the principal directions, although the theory is readily expanded to the more general case.22 In the reference state, the gel is a unit cube of a dry polymer, containing no solvent and subject to no applied forces (Fig. 1a). In the current state, submerged in a solution and subject to applied forces, the gel absorbs solvent molecules and various ionic species, and stretches into a rectangular block of dimensions l1, l2 and l3 (Fig. 1a). The volume ratio of gel to dry polymer, J, known as the swelling ratio, relates to the dimensions of the rectangular block by J ¼ l1l2l3. The gel and the external solution contain four mobile species: solvent molecules, protons, counterions with charge opposite to the charge xed on the polymer chains, and co-ions with charge of the same sign as the xed charge (Fig. 1b). We dene the nominal concentration Ca of species a as the number of species a in the current state divided by the volume of the dry network. The same number divided by the volume of the gel in the current state denes the true concentration ca. The two quantities are evidently related by Ca ¼ ca J. As part of the swelling process, the acidic groups AH on the polymer chains of the network may dissociate, leaving xed charge A on the chains. This dissociation reaction, AH 4 A + H+, reaches equilibrium when CHþ CA Na Ka ¼ CAH J
(1)
where Na is Avogadro's constant and Ka is the acid dissociation constant.
Fig. 1 (a) A unit block of the polyelectrolyte gel is submerged in a solution environment containing various ionic species. The inset is the structural formula of polyacrylamide-co-acrylic acid. (b) The polyelectrolyte gel, composed of cross-linked polymer chains with fixed charge and acidic groups, absorbs solvent and mobile ions, while the external solution contains solvent, protons, co-ions and counterions.
Soft Matter
We recall the two basic assumptions underlying the concept of an ideal elastomeric gel. (1) Molecular incompressibility: the volume of the ideal elastomeric gel is equal to the sum of the volume of the dry network and that of the solvent, J ¼ 1 + UsCs,
(2)
where Us is the volume per solvent molecule s. In the case of polyelectrolyte gels, we assume further that the concentrations of the ionic species are generally much lower than those of the solvent species,17 so that eqn (2) remains valid. (2) Separability of the Helmholtz free energy: the Helmholtz free energy of the ideal elastomeric gel consists of one term that represents the elastic stretching of the polymer network and one term that is associated with mixing of the polymer and solvent. In the case of polyelectrolyte gels, the xed and mobile ions also contribute to the Helmholtz free energy. Following prior work,12,16,17 we formulate the total Helmholtz free energy referred to a unit volume in the reference state as the sum of the following contributions: W ¼ Wstretch + Wmix + Wion + Wdis,
(3)
where Wstretch is the energy contribution due to the stretching of the network. This term is a function of the stretches (l1, l2, l3) and depends on the cross-link density of the network. The term Wmix is the contribution associated with the mixing of the polymer and solvent. This term is a function of the swelling ratio, J, given by eqn (2) and is taken independent of the crosslink density. The term Wion is associated with the mixing of mobile ions and generally takes the form Wion ¼ Wion(J, CH+, C+, C), where the subscripts + and – refer to the counterions and co-ions, respectively. If the concentrations of the ions are low and the polymer network is weakly charged, the electrostatic interactions between the ions in the gel are negligible,12 and Wion is determined mainly by the entropy of mixing of the mobile ions: ! " ! C þ Cþ Wion ¼ kT CHþ log refH 1 þ Cþ log ref 1 cþ J cHþ J # C þ C log ref 1 ; (4) c J where cref refers to the standard concentration, dened as 1 mol L1. For strongly charged polyelectrolyte gels, the energy contribution due to electrostatic interactions between mobile ions and xed charge may be signicant and needs to be considered.11,23 The electrostatic effect can be readily applied to this theoretical framework, but is not further considered in this study. The last term in eqn (3), Wdis, denotes the change in free energy caused by dissociation of the acidic groups on the polymer chains. This term contains both an entropic and an enthalpic term, CA Wdis ¼ kT CA log CA þ CAH CAH þ gCA ; (5) þ CAH log CA þ CAH
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where g is the increase in enthalpy when an acidic group dissociates. Under the constraints of mass conservation and electroneutrality, CA and CAH are readily expressed in terms of CH+, C+ and C: CA ¼ CH+ + C+ C
(6)
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and CAH ¼ f/v (CH+ + C+ C),
(7)
where f is the number of acidic groups attached to the network divided by the total number of monomers forming the network and v is the volume per monomer. Consequently, the total Helmholtz free energy can be written as a function of six independent variables: the stretches (l1, l2, l3) and the concentrations of the protons, counterions, and co-ions (CH+, C, C+). When a dry polymer network is submerged in a solution and subjected to forces, the network absorbs the mobile species and deforms, eventually attaining a state of thermodynamic equilibrium. The condition for equilibrium can be formulated as follows. We dene the stresses s1, s2 and s3 as the applied forces divided by the areas of the faces of the block of the gel in the current state (Fig. 1a). Thus, the forces on the faces of the rectangular block are s1l2l3, s2l3l1 and s3l1l2. Associated with a small change in the dimensions of the gel, these forces perform work on the gel s1l2l3dl1 + s2l3l1dl2 + s3l1l2dl3. The gel and the solution can also exchange four mobile species: solvent molecules, protons, counterions and co-ions. Let ma be the chemical potential of species a—since the gel and the solution are in a state of equilibrium, the chemical potential of each species is uniform in the system. When a number dCa (per unit volume of dry gel) of species a are transferred from the solution to the gel, the free energy of the gel changes by madCa. This statement holds for solvent molecules, counterions, and co-ions but not for protons because the number of protons in the gel can also change by dissociation of the acidic groups on the polymer chains. Consequently, in equilibrium the total Helmholtz free energy W obeys dW ¼ s1l2l3dl1 + s2l3l1dl2 + s3l1l2dl3 + msdCs + mH+(dCH+ dCA) + mdC + m+dC+. Combining eqn (2)–(8) yields the following expression, " # vWstretch ðs1 þ Pmix þ Pion Þl2 l3 dl1 vl1 " # vWstretch þ ðs2 þ Pmix þ Pion Þl3 l1 dl2 vl2 " # vWstretch þ ðs3 þ Pmix þ Pion Þl1 l2 dl3 vl3 " # vW þ ðmþ mHþ Þ dCþ vCþ " # vW vW þ ðm þ mHþ Þ dC þ dCHþ ¼ 0 vC vCHþ
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(8)
(9)
where dWmix ðJÞ ; dJ
(10)
vWion ðJ; CHþ ; Cþ ; C Þ ms þ : vJ Us
(11)
Pmix ¼ Pion ¼
here Pmix is the osmotic pressure in the gel in equilibrium with the pure solvent, while Pion represents the osmotic pressure in the gel due to the mobile ions; ms/Us is the osmotic pressure in the gel due to the chemical potential of the solvent in the external solution containing ionic species: ms ¼ kTUs c*Hþ þ c*þ þ c* (12) where c*a is the true concentration of species a in the external solution.17 Substituting eqn (4) and (12) into (11) yields (13) Pion ¼ kT cHþ þ cþ þ c c*Hþ c*þ c* : The network, the solvent and the applied forces are in equilibrium, if eqn (9) holds for arbitrary and independent small changes in the three stretches (l1, l2 and l3) and the three nominal concentrations (C+, C, CH+). Consequently, the expressions in the brackets in front of each of the six terms in eqn (9) must vanish individually, yielding the following equations: vWstretch s1 ¼ Pmix Pion ; (14a) l2 l3 vl1 s2 ¼
vWstretch Pmix Pion ; l1 l3 vl2
(14b)
s3 ¼
vWstretch Pmix Pion : l1 l2 vl3
(14c)
vW ¼ mþ mHþ : vCþ
(15a)
vW ¼ m þ mHþ : vC
(15b)
vW ¼ 0: vCHþ
(16)
The applied stresses are balanced by the elasticity of the network, the osmotic pressure due to mixing of the polymer and solvent, and the osmotic pressure due to mixing of ions. Eqn (14) expresses mechanical equilibrium; eqn (15) provides the equilibrium conditions with respect to the exchange of counterions and co-ions between the gel and its surrounding; eqn (16) formulates the equilibrium condition for dissociation of the acidic groups on the polymer chains. Substituting eqn (3)– (5) into (16) leads to g NA Ka ¼ cref : (17) Hþ exp kT Eqn (2), (14)–(16) constitute the equations of state of the polyelectrolyte gel, relating the thermodynamic variables: l1, l2, l3, s1, s2, s3, C+, C, CH+. Once the free energy contributions due to stretching of the network, Wstretch, and the osmotic pressure,
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Pion, have been characterized experimentally, the equations of state provide a complete description of the chemo-mechanical behavior of the polyelectrolyte gel. The energy contribution due to stretching is readily determined from stress–strain measurements. Characterizing Pion is less straightforward since it requires knowledge of the concentrations (CH+, C+, C) inside the gel, which are difficult to measure in practice.24 Instead, we adopt Donnan's theory,17 which provides a relationship between the ion concentrations inside the gel and those in the external solution, 1 cþ cHþ c ¼ ¼ * : (18) c*þ c*Hþ c Tanaka et al. successfully applied Donnan's theory to capture the swelling behavior of an ionic gel quantitatively.12 The concentrations of the three species inside the gel also satisfy the equilibrium condition (1) for dissociation of the acidic groups, yielding cHþ ðcHþ þ cþ c Þ ¼ NA K a : f ðcHþ þ cþ c Þ vJ
(19)
If f, J, c*Hþ , c*+, c*, and Ka are known, the concentrations (cH+, c+, c) of the ionic species in the gel are solved readily from eqn (18) and (19).
3.
Experimental methods
To determine the free energy contributions of stretching and the osmotic pressure in the gels in equilibrium with the pure solvent, and to verify the equations of state for polyelectrolyte gels, a series of polyacrylamide-co-acrylic acid gels were synthesized according to the following protocol;12 a summary of the composition and designation of each sample is provided in Table 1. Acrylamide (AAM), acrylic acid (AA), N,N0 -methylenebis(acrylamide) (MBAA) and ammonium persulfate (APS) were acquired from Sigma Aldrich. The overall monomer (AAM + AA) concentration was 1000 mM. As indicated in Table 1, the mole fraction of acrylic acid, f, was varied systematically, but kept small to ensure a low density of acidic groups on the polymer chains. Different cross-link densities were obtained by varying the weight percentage of MBAA (Table 1). APS (7 mM) was added as an initiator for the free-radical polymerization. The chemicals (AAM, AA, MBAA and APS) were mixed using 10 mL syringes. Part of the mixture (6 mL) was poured into a glass mold of 1 mm thickness. The rest of the solution (4 mL) was kept in the syringe to fabricate samples with a cylindrical shape. Gelation occurred over a period of one hour on a hot plate at 70 C. To prevent hydrolysis of acrylamide and to precisely control the fraction of acidic groups, we did not use the usual accelerator, tetramethyl-ethylenediamine (TEMED). Aer gelation, the samples were stored at room temperature for 24 hours, and then transferred to a large body of distilled water to remove unreactive chemicals and other impurities, while the gels were allowed to swell freely to equilibrium. This step lasted for
Soft Matter
Table 1 Composition and parameters of polyacrylamide-co-acrylic acid gelsa
Notation
MBAA (wt%)
f
J0
NkT (kPa)
AA01-080 AA02-080 AA04-080 AA01-216 AA02-216 AA04-216 AA01-433 AA02-433 AA04-433
0.80 0.80 0.80 2.16 2.16 2.16 4.33 4.33 4.33
0.01 0.02 0.04 0.01 0.02 0.04 0.01 0.02 0.04
140 590 654 71 79 88 36 37 61
2.33 3.62 3.71 10.83 10.22 10.07 15.96 22.22 17.39
a f is the mole fraction of AA versus (AA+AAM), J0 is the equilibrium swelling ratio of the gel in distilled water, and NkT is the crosslink density.
around 10 days, during which fresh distilled water was added every 24 hours. The mass of the fully swollen cylindrical samples (mgel) was measured using an analytical scale with an accuracy of 10 mg. The samples were subsequently frozen at a temperature of 80 C and transferred to a freeze–dry system (Labconco Corporation) with a collector temperature of 50 C and a vapor pressure of 0.040 mbar. This dehydration process took three days to complete, aer which the mass of the dehydrated samples (mdry) was measured. Swelling ratios were then calculated using
mgel mdry rwater J ¼1þ : (20) mdry =rdry where rwater is the density of water taken as 1.000 g cm3 and rdry is the density of the dry polymer network taken as the density of polyacrylamide at 1.443 g cm3.25 The swelling ratios, J0, in distilled water are listed in Table 1 for each of the samples. Swelling ratios in subsequent experiments were calculated from J0 and the change in diameter, J ¼ J0(Df/Di)3, where Di and Df are the initial and nal diameters of the sample. The responses of the gels to changes in pH and salinity of the external solution were characterized using samples AA04-080 and AA04-433. Small gel cylinders of diameters 8 or 12 mm were punched out of fully swollen gel sheets. These samples were transferred to solutions of pH varying from 2.5 to 12. Acidic solutions (pH < 7) were prepared by adding HCl to distilled water, while KOH was used for alkaline solutions (pH > 7). Gel cylinders were also transferred to solutions with NaCl concentrations varying from 107 M to 102 M. The use of buffer solutions was avoided to not introduce additional electrochemically active species. The external solutions were renewed every 24 h. We recorded the diameters of the cylinders using a pair of calipers before and aer reaching the new equilibrium state. Small gel cylinders with 8 mm diameter were used for the mechanical tests. A series of uniaxial compression tests were performed using an AR-G2 rheometer (TA Instruments). The rheometer platen approached the samples at a speed of 5 mm This journal is © The Royal Society of Chemistry 2014
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s1 until a rise in force was detected, indicating contact between the platen and the gel. Upon the contact, the displacement rate of the platen was increased to 20 mm s1 until a pre-determined stretch ratio, typically 0.8–0.9, was achieved. The entire loading process was nished within 2 minutes to avoid any redistribution of solvent inside the gel and thus maintain incompressibility. Both force and displacement were recorded continuously throughout the experiment.
4. Results and discussion Mechanical characterization Fig. 2a shows the swelling ratios in distilled water for each of the samples. Typical values are well in excess of 50 so that the water content of the most swollen gels is greater than 95%. As illustrated in the gure, the swelling ratio changes rapidly with the fraction of acidic groups and the cross-link density. Even in distilled water, weakly charged and loosely cross-linked gels (e.g. AA02-080 and AA04-080) can swell to more than 500 times their dry volume. The contribution of the free energy of stretching, Wstretch, was determined from uniaxial compression tests on samples fully swollen in distilled water. During the tests, the gels behaved like incompressible solids (l1l2l3 ¼ l03), because there was insufficient time for diffusion. We then used the Gaussian chain model, " 2 # 2 NkT l3 l1 s3 s1 ¼ ; (21a) l0 l0 l0 NkT s3 s2 ¼ l0
" 2 # 2 l3 l2 ; l0 l0
(21b)
to t the experimental stress–stretch relations, where l0 ¼ J01/3 is the stretch ratio aer swelling in distilled water, N is the number of polymer chains per unit volume of the dry network, k is Boltzmann's constant, and T is the absolute temperature. Considering uniaxial compression along the x3-direction, these equations reduce to a single equation NkT 2 1 l ; s3 ¼ (22) l0 l
where l ¼ l3/l0 is the stretch relative to the freely swollen state. It is evident from eqn (22) that NkT/l0 represents the shear modulus of the gels, while also allowing us to determine the cross-link density (N) from the initial slope of the stress–stretch curve. In our previous work on polyacrylamide gels, the Gaussian-chain model provided an excellent description of the mechanical behavior of the gels. As expected, the Gaussianchain model also provides a very good t for the experimental stress–stretch curves for the polyacrylamide-co-acrylic acid gels. Since the gels were swollen in distilled water, they had moderate swelling ratios and did not exhibit any non-Gaussian behavior. Fig. 2b illustrates that NkT increases with increased amounts of cross-linker, but is independent of the fraction of acrylic acid, f. A correlation between the modulus and the ionic components inside the gel has been reported in the literature,13 a phenomenon attributed to the electrostatic interaction between the xed ions on the polymer chain. In this study, the fraction of acidic groups is intentionally maintained at a level below 5%, so that xed ions are on average separated by at least 20 neutral segments resulting in a weak electrostatic interaction between the ions.
Characterization of the osmotic pressure terms The osmotic pressure Pion of the gel samples can be calculated by combining eqn (13) with eqn (18) and (19). The results for an external solution of distilled water with a slightly acidic pH of 6.5 are plotted as a function of f/J in Fig. 3a – data for gels with different cross-link densities and acid group contents are shown, but they collapse into a single master curve because the ionic osmotic pressure is a unique function of the current concentration of acid groups, f/J, and the pH of the external solution. The ionic osmotic pressure is shown to increase with acid group content and decrease with decreasing cross-link density. The former is caused by the higher concentration of active species and the latter occurs because the equilibrium swelling ratio is larger, which results in a lower concentration of active species. Using the equations of state, it is now possible to derive the function Pmix(J) from our experiments without assuming a specic model for Wmix. The stress–strain curves suggest that Wstretch is well described by the Gaussian-chain model, i.e.,
Fig. 2 (a) The equilibrium swelling ratio of the gels fully swollen in distilled water (J0) and (b) the crosslink density (NkT) of the gels, obtained from the stress–stretch measurements, is plotted versus the fraction of acidic groups (f) and the crosslinker input (MBAA%).
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Fig. 3 (a) The calculated osmotic pressure due to ions (Pion) versus the density of acidic groups (f/J), and (b) the osmotic pressure due to mixing (Pmix) as a function of the swelling ratio (J), measured from gels fully swollen in distilled water. The same color implies the same input of crosslinker (MBAA) in the synthesis, while the fraction of acidic groups (f) is varied. The data points marked by PAAM data were extracted from the measurement on polyacrylamide gels.21
1 Wstretch ¼ NkT l1 2 þ l2 2 þ l3 2 3 2 logðl1 l2 l3 Þ : 2
(23)
Under free-swelling conditions, the stretches are isotropic, l1 ¼ l2 ¼ l3 ¼ J01/3 and all applied stress components vanish, s1 ¼ s2 ¼ s3 ¼ 0. Substituting into eqn (14) yields NkT 2 Pmix ¼ J0 3 1 Pion ; (24) J0 from which Pmix can be obtained directly once J0, NkT, and Pion have been determined experimentally. Fig. 3b shows the plot of the function, Pmix, as a function of swelling ratio for gels with different cross-link densities and acid contents. We have also included Pmix measured for polyacrylamide (PAAM) hydrogels in our previous work.21 Regardless of the hydrogel cross-link density or acid content, Pmix collapses into a single master curve over a large range of swelling ratios, i.e., Pmix is a function of a single variable, J. This is a direct consequence of the low concentration of acid groups in the hydrogels, which makes the overall chemistry of the polymer chains close to that of pure polyacrylamide. The responses of the gels to changes in pH or salinity of the external solution were also characterized. Fig. 4a shows the plot of the swelling ratio as a function of pH, along with experimental data obtained by Tanaka et al.12 A qualitative understanding of the trends in the data is as follows. When the pH of
the external solution is low, protons are abundant and the acidic groups on the polymer chains are fully protonated. The gel behaves like a neutral gel with a relatively small swelling ratio because of the absence of an ionic osmotic pressure Pion. As the pH approaches 8, the acidic groups dissociate and create xed charge on the polymer chains. An imbalance of mobile ions builds up between the gel and the external solution, and the increasing Pion makes the gel swell. Once the pH increases further, the acidic groups are fully dissociated and the ionic osmotic pressure Pion drops because of the abundant presence of mobile ions everywhere. The response to the changes in salinity, the concentration of NaCl in the external solution, is shown in Fig. 4b; an understanding similar to that for Fig. 4a applies here. The gel acts as an ion exchanger. As the concentration of NaCl increases, protons in the gel are exchanged with Na+ ions and the acidic groups continue to dissociate. At approximately 104 M of NaCl, the acidic groups are completely dissociated. The excess of NaCl reduces the ion imbalance between the gel and the solution, and the degree of swelling decreases. In the model we assumed an ideal ionic solution and insignicant effects due to electrostatic interactions between xed and mobile ions, both valid for low concentrations (
